Rank-one measurements limit feasible sets for low-rank PSD matrices.
problem Feasibility of PSD matrices under rank-one measurements.
method Characterization of feasible sets for PSD matrices given rank-one projections.
result Radius of feasible sets determines singleton solution sets for low-rank matrices.
This paper tackles fitting multilevel low rank matrices by addressing three problems.
problem Fitting a given matrix by an MLR matrix in the Frobenius norm.
method Factor fitting, rank allocation, and hierarchical partitioning.
result The proposed methods can fit a given matrix by an MLR matrix in the Frobenius norm.
New algorithm learns low-rank matrices with linear number of samples.
problem Learning low-rank matrices efficiently in latent-variable applications.
method Proposed algorithm that uses linear number of samples in high dimension.
result Learning kimesk, rank-r, matrices requires $Ω( rac{kr}{ε^2})$ samples. Algorithm recovers multiple low-rank matrices from unlabeled data.
problem Learning mixtures of low-rank models from unlabelled data.
method Three-stage meta-algorithm that copes with non-convexity and noise.
result Near-optimal sample and computational complexities under Gaussian designs.
Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure ap…
New framework solves low-rank optimization problems to certifiable optimality.
problem Low-rank optimization problems with certifiable solutions.
method Mixed-Projection Conic Optimization framework using symmetric projection matrices and outer-approximation algorithms.
result Solves low-rank problems to certifiable optimality, outperforming existing methods.
Develops PRPCA for smooth image recovery combining low-rank and smoothness.
problem Image matrix recovery under low-rank and smoothness assumptions.
method Projected Robust PCA (PRPCA) framework combining low-rank and smoothness.
result Explicit statistical guarantees for PRPCA, reducing matrix dimensionality.
Paper proposes a method for estimating complex low-rank matrices from phase-only measurements.
problem Estimating complex low-rank matrices from magnitude-only measurements.
method A hierarchical prior model with a Gaussian-Wishart distribution is used to promote low-rankness. A variational EM algorithm is developed to solve the problem.
result The proposed method is less sensitive to initialization and performs well with random initialization.
We address some theoretical guarantees for Schatten-p quasi-norm minimization (p∈(0,1]) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measurement operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition…
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
Weight Decay induces low-rank weight matrices in neural networks, improving generalization.
problem Improving generalization in neural networks.
method Training ReLU NN with Weight Decay and Stochastic Gradient Descent.
result The weight matrix of a trained NN is approximately rank-two.
Proposes a new method for high-dimensional data analysis.
problem Sparse PCA limitations in high-dimensional data analysis.
method Low-rank principal eigenmatrix analysis, matricized rank-truncated power method.
result Competitive empirical performance in synthetic data sets.
RES-PCA efficiently recovers low-rank matrices without precise rank knowledge.
problem Inefficient and computationally expensive RPCA methods.
method RES-PCA, a scalable and linearly efficient RPCA method.
result RES-PCA is faster and more robust than existing scalable methods.
LOCUS separates brain network connectivity matrices efficiently.
problem High dimensionality, latent sources, and spurious findings in analyzing brain connectivity matrices.
method LOCUS: low-rank structure with uniform sparsity, iterative Node-Rotation algorithm.
result LOCUS achieves more efficient and accurate source separation for connectivity matrices.
Extends MMF to nonsymmetric matrices for hierarchical structure.
problem Capturing hierarchical structure in nonsymmetric matrices.
method Multiresolution Matrix Factorization (MMF) extended to nonsymmetric matrices.
result Effective for matrix compression tasks, outperforming low-rank methods.
We develop latent variable models for Bayesian learning based low-rank matrix completion and reconstruction from linear measurements. For under-determined systems, the developed methods are shown to reconstruct low-rank matrices when neither the rank nor the noise power is known a-priori. We derive relations between th…
Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on ma…
CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrice…
Matrix completion is a problem that arises in many data-analysis settings where the input consists of a partially-observed matrix (e.g., recommender systems, traffic matrix analysis etc.). Classical approaches to matrix completion assume that the input partially-observed matrix is low rank. The success of these methods…
Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best …
New method corrects quantization errors in LLMs using low-rank matrices.
problem Correcting quantization errors in large language models.
method Introducing low-rank weight matrices to correct quantized activations in LLMs.
result Reduces accuracy gap with original model by more than 50% using low-rank matrices.
TensorGuide improves LoRA efficiency and expressivity through joint tensor-train optimization.
problem Limited expressivity and generalization of standard LoRA.
method TensorGuide uses a unified tensor-train structure with controlled Gaussian noise to generate correlated low-rank matrices.
result TensorGuide achieves superior accuracy and scalability with fewer parameters compared to standard LoRA and TT-LoRA.
Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability issue arises with tra…
This paper uncovers the low-rank structure of neural network Hessians.
problem Understanding the structure of Hessians in neural networks.
method Proposes a decoupling conjecture to decompose layer-wise Hessians into Kronecker products of smaller matrices.
result Proves the structure of top eigenspaces in 2-layer networks and shows high overlap in top eigenvectors across different models.
New algorithms improve RPCA for large matrices with upper rank bounds.
problem Efficiently decompose large matrices into low-rank and sparse parts.
method Combine regularization and matrix multiplication approaches with upper rank bounds.
result Proposed algorithms are faster and more robust than existing methods.
Mirror descent algorithm recovers low-rank matrices in matrix sensing.
problem Matrix sensing with low-rank matrices under certain conditions.
method Discrete-time mirror descent applied to empirical risk with Bregman divergence analysis.
result Mirror descent converges to a matrix minimizing a specific nuclear norm-related quantity.
We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left an…
Efficient algorithm for Hadamard decomposition of matrices.
problem Decomposing matrices into low-rank factors efficiently.
method Alternating optimization with SVD-inspired initialization and momentum.
result Significantly improved performance compared to existing methods.
Paper finds a lower bound for estimating low-rank matrices in logistic regression.
problem Estimating low-rank coefficient matrices in logistic regression.
method Derives a minimax lower bound on the risk.
result The bound depends on matrix dimensions, rank, and sample size.
New method improves matrix completion accuracy, especially in noisy data.
problem Noisy matrix completion in recommendation systems and signal processing.
method Residual Spectral Matching criterion and pseudo-gradient algorithms.
result Improved numerical performance in noisy data environments.
ScaledGD improves gradient descent for ill-conditioned low-rank matrix estimation.
problem Efficiently solving ill-conditioned low-rank matrix estimation problems.
method Scaled Gradient Descent (ScaledGD) with adaptive pre-conditioners.
result Linear convergence rate independent of condition number, low per-iteration cost.
In this letter, we propose an algorithm for recovery of sparse and low rank components of matrices using an iterative method with adaptive thresholding. In each iteration, the low rank and sparse components are obtained using a thresholding operator. This algorithm is fast and can be implemented easily. We compare it w…
Paper proposes a new algorithm for graph learning with covariance constraints.
problem Graphical models and factor analysis not jointly leveraged in graph learning processes.
method Penalized maximum likelihood estimation of an elliptical distribution with Riemannian optimization.
result Effectiveness of the proposed approach demonstrated on real-world data sets.
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
Algorithm learns a better sketch matrix for low-rank approximations.
problem Efficiently compute low-rank approximations of large matrices.
method Uses a learned sketch matrix instead of random matrix for optimization.
result Learned sketch matrix reduces approximation loss significantly compared to random matrix.
Study evaluates thresholds for removing noise from DNN weights using random matrix theory.
problem Removing noise from deep neural network weights for better approximation.
method Model weights as signal + noise, use random matrix theory to estimate thresholds, evaluate using cosine similarity.
result Proposed threshold estimation method improves approximation quality.
Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in {\it quantum state tomography} (QST) is to estimate the unknown l…
Paper proposes a new method to separate low rank and sparse matrices without bias.
problem Recovering low rank and sparse matrices from measurements.
method Uses nonconvex regularizers and alternating proximal gradient descent.
result Error bounds for the algorithm applied to sparse optimization, matrix completion, and robust PCA.
Algorithm compresses large matrices by approximating them as low rank and low precision factors.
problem Efficiently storing and processing large matrices with billions of elements.
method Randomized sketching and quantization of matrix columns to achieve low rank and low precision factorization.
result Achieves compression ratios as low as one bit per matrix coordinate while maintaining or improving performance.
A new method reduces high-dimensional filtering to quadratic complexity.
problem High-dimensional dynamical systems inference and simulation.
method Low-rank Kalman filtering using dynamical low-rank integrator.
result The method reproduces exact Kalman filter in low-rank limit.
Efficiently implements MEG for low-rank matrix optimization problems.
problem Optimization over spectrahedron with low-rank matrices.
method Matrix Exponentiated Gradient (MEG) method with efficient implementations.
result Methods converge from a warm-start initialization with similar rates to full-SVD-based counterparts.
Polynomial sketch approximates functions of low-rank matrices efficiently.
problem Approximating element-wise functions of low-rank matrices without full access.
method Combining polynomial approximation and tensor sketch for monomials.
result Efficient algorithm with lower complexity than full matrix access.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…
In the paper, we consider the problem of link prediction in time-evolving graphs. We assume that certain graph features, such as the node degree, follow a vector autoregressive (VAR) model and we propose to use this information to improve the accuracy of prediction. Our strategy involves a joint optimization procedure …
Geometric families of low-rank covariances improve flexibility and tractability in high dimensions.
problem Interpolating and identifying covariance matrices in high dimensions with limited data.
method Differential geometric construction of low-rank covariance families, interpolation on manifolds, and distance minimization for identification.
result Differential geometric covariance families offer significant flexibility and computational tractability.
Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is…
Flora uses random projections to achieve high-rank updates with low memory usage.
problem Excessive memory usage in large neural networks during training.
method Flora approximates LoRA using random projections to enable high-rank updates with sublinear space complexity.
result Flora achieves high-rank updates with significantly reduced memory usage compared to LoRA.
Unified model for tensor completion using low-rank and sparse Tucker decomposition.
problem Estimating missing data from incomplete tensor measurements.
method Unified low-rank and sparse enhanced Tucker decomposition model with ADMM.
result Our model achieves higher recovery accuracy on various real-world data sets.